cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A130379 Least common multiple of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutation A130369/A130370.

Original entry on oeis.org

1, 1, 2, 5, 24, 35, 17304, 105210, 15002667388800, 2803962610087320, 390995845903819693817280, 4427769139935736194600, 426341479886584667397117049422960
Offset: 0

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Author

Antti Karttunen, Jun 05 2007

Keywords

Comments

Note the non-monotone drop from a(10) to a(11).

A130967 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutations A130919/A130920.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 66, 189, 574, 1773, 5640, 18208, 59774, 198654
Offset: 0

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Author

Antti Karttunen, Jun 11 2007

Keywords

Crossrefs

Cf. A130968, A057513.

A130968 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutations A130919/A130920.

Original entry on oeis.org

1, 1, 2, 3, 7, 10, 25, 54, 131, 331, 864, 2292, 6169, 16835
Offset: 0

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Author

Antti Karttunen, Jun 11 2007

Keywords

Crossrefs

Cf. A130967, A057546.

A137993 A014138 (= partial sums of Catalan numbers starting with 1,2,5) mod 3.

Original entry on oeis.org

1, 0, 2, 1, 1, 1, 1, 0, 2, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 2, 0, 1, 1, 1, 1, 2, 0, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0
Offset: 0

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Author

M. F. Hasler, Mar 16 2008

Keywords

Comments

As usual, "mod 3" means to choose the unique representative in { 0,1,2 } of the equivalence class modulo 3Z.
Here the conventions of A014138 are used, but it seems somehow unnatural to start with offset 0 corresponding to the Catalan number A000108(1).
For m>1, the length of the m-th block of nonzero elements (and thus the approximate length of the m-th string of consecutive 1's) is given by 2 A137822(m)-1.

Crossrefs

Cf. A014138, A000108, A137821-A137824, A107755, A137992, A014137(n+1) = a(n)+1 (mod 3).

Programs

  • PARI
    A137993(n) = lift( sum( k=1,n+1, binomial( 2*k,k )/(k+1), Mod(0,3) ))

Formula

a(n) = sum( k=1..n+1, C(k) ) (mod 3), where C(k) = binomial(2k,k)/(k+1) = A000108(k).
a(n) = 0 <=> n+1 = 2 A137821(m) for some m.

A244317 n occurs A014138(n) times.

Original entry on oeis.org

0, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6
Offset: 0

Views

Author

Antti Karttunen, Jul 18 2014

Keywords

Comments

For n >= 1, a(n) = 1 + the least k such that A014143(k) >= n.
Useful when computing A244314.

Crossrefs

Cf. A014138, A014143, A244314, A072643, A081288, A244160.

Programs

  • Mathematica
    Join[{0},Flatten[Table[#[[2]],#[[1]]]&/@With[{nn=6},Thread[{Join[Accumulate[ CatalanNumber[ Range[ nn]]]],Range[nn]}]]]] (* Harvey P. Dale, Sep 06 2023 *)
  • Scheme
    (define (A244317 n) (if (zero? n) n (let loop ((k 0)) (if (>= (A014143 k) n) (+ 1 k) (loop (+ 1 k))))))

Formula

For all n >= 0, a(A014143(n)) = n+1 and a(1+A014143(n)) = n+2.

A081149 Number of odd cycles in range [A014137(2n)..A014138(2n)] of permutation A057505/A057506.

Original entry on oeis.org

1, 1, 4, 9, 24, 62, 162, 447, 1234
Offset: 0

Views

Author

Wouter Meeussen and Antti Karttunen, Mar 10 2003

Keywords

Crossrefs

Cf. A014137, A014138, A057505, A057506, A081148.

Formula

a(n) = A081148(2n+1).

A081154 Number of odd cycles in range [A014137(2n)..A014138(2n)] of permutation A057505/A057506, with no fixed points of either A057163 or A057164.

Original entry on oeis.org

0, 0, 2, 6, 18, 50, 142, 388, 1114
Offset: 0

Views

Author

Wouter Meeussen and Antti Karttunen, Mar 10 2003

Keywords

Crossrefs

Cf. A081153, A014137, A014138, A057505, A057506, A057163, A057164.

Formula

a(n) = A081153(2n+1).

A081167 Number of cycles in range [A014137(2n)..A014138(2n)] of permutation A057505/A057506.

Original entry on oeis.org

1, 2, 10, 46, 236, 1288, 6984, 38528, 216648, 1240818
Offset: 0

Views

Author

Wouter Meeussen and Antti Karttunen, Mar 10 2003

Keywords

Crossrefs

Cf. A057507, A081151, A014137, A014138, A057505, A057506.

Formula

a(n) = A057507(2n+1).

A089417 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A082333/A082334.

Original entry on oeis.org

1, 1, 1, 2, 7, 6, 10, 20, 37, 70, 130, 272, 480, 954, 1750, 3462, 6481, 12922, 24372, 48702, 92490
Offset: 0

Views

Author

Antti Karttunen, Nov 29 2003, Proposed by Wouter Meeussen, Spring 2003

Keywords

Comments

The number of orbits to which the corresponding automorphism(s) partitions the set of A000108(n) binary trees with n internal nodes.
Is the non-monotone notch from a(4)=7 to a(5)=6 the only one?

Links

Crossrefs

Cf. A089411.

A089429 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A082339/A082340.

Original entry on oeis.org

1, 1, 2, 4, 10, 26, 74, 218, 672, 2134, 6964, 23206, 78724, 271014, 944766, 3328872, 11838378, 42441326, 153236900, 556746772, 2034098988
Offset: 0

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

The number of orbits to which the corresponding automorphism(s) partitions the set of A000108(n) binary trees with n internal nodes.

Links

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